MATHEMATICAL COMMUNICATIONS Math. Commun., Vol. 15, No. 2, pp. 303-310 (2010)
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Fuzzy bases and the fuzzy dimension of fuzzy vector spaces∗ Fu-Gui Shi1,† and Chun-E Huang2 1
Department of Mathematics, School of Science, Beijing Institute of Technology, Beijing-100 081, P. R. China 2 Department of Thermal Engineering, Tsinghua University, Beijing-100 081, P. R. China
Received February 15, 2009; accepted November 10, 2009
Abstract. In this paper, new definitions of a fuzzy basis and a fuzzy dimension for a fuzzy vector space are presented. A fuzzy basis for a fuzzy vector space (E, µ) is a fuzzy set β on E. The cardinality of a fuzzy basis β is called the fuzzy dimension of (E, µ). The fuzzy dimension of a finite dimensional fuzzy vector space is a fuzzy natural number. For e1 and E e2 are a fuzzy vector space, any two fuzzy bases have the same cardinality. If E e1 + E e2 ) + dim(E e1 ∩ E e2 ) = dim(E e1 ) + dim(E e2 ) and two fuzzy vector spaces, then dim(E f ) + dim(imf e hold without any restricted conditions. f ) = dim(E) dim(kerf AMS subject classifications: 03E72, 08A72 Key words: Fuzzy vector space, fuzzy basis, fuzzy dimension
1. Introduction After Katsaras and Liu [4] introduced the concept of fuzzy vector spaces, many scholars investigated its properties and characteristics (see [1, 2, 5, 6, 7], etc). For a fuzzy vector space, Lowen [5] gave the definition of fuzzy linear independence which is a direct generalization of normal linear independence. Subsequently P. Lubczonok [7] introduced another alternative concept of fuzzy linear independence, and introduced a definition of fuzzy bases which are the bases of corresponding vector space and satisfy fuzzy linear independence. In this case, the fuzzy bases of fuzzy vector space are still crisp. At the same time, a concept of dimension for a fuzzy vector space was introduced. Lowen [5] defined a dimension of the n-dimensional Euclidean space as an n-tuple. P. Lubczonok [7] defined a fuzzy dimension for all fuzzy vector spaces as a non-negative real number or infinity, and proved that for e1 and E e2 the statement finite dimensional vector spaces E e1 + E e2 ) + dim(E e1 ∩ E e2 ) = dim(E e1 ) + dim(E e2 ) dim(E
(1)
is true under certain conditions. In this paper, we redefine fuzzy basis of a fuzzy vector space (E, µ) as a fuzzy set β on E and discuss its properties. Subsequently, by the definition of fuzzy basis, ∗ This
work was supported by the National Natural Science Foundation of China, No. 10971242. author. Email addresses:
[email protected] (F. -G. Shi),
[email protected] (C. -E Huang)
† Corresponding
http://www.mathos.hr/mc
c °2010 Department of Mathematics, University of Osijek
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F. -G. Shi and C. -E Huang
we redefine fuzzy dimension of (E, µ), it is the cardinality of a fuzzy basis β. The fuzzy dimension of a finite dimensional fuzzy vector space is a fuzzy natural number. For a fuzzy vector space, any two fuzzy bases have the same cardinality. We also f ) = dim(E) e hold without any f ) + dim(kerf prove that the formulas (1) and dim(imf restricted conditions.
2. Preliminaries For a set X, a fuzzy set A on X is a mapping A : X → [0, 1]. The set of all fuzzy sets on X are denoted by [0, 1]X . For A ∈ [0, 1]X and a ∈ [0, 1], define A[a] = {x ∈ X : A(x) > a}, A(a) = {x ∈ X : A(x) > a}. We often do not distinguish a crisp subset A of X and its characteristic function χA . The cardinality of a fuzzy set was introduced in [11] and the notion of fuzzy natural integers was also investigated and developed in [3, 8]. In [10], they were generalized to an L-fuzzy setting as follows: Definition 1 (see [10]). Let N denote the set of all natural numbers and let L be a complete lattice. An L-fuzzy natural integer is an antitone mapping λ : N → L satisfying ^ λ(0) = >L , λ(n) = ⊥L , n∈N
where >L and ⊥L are the largest element and the smallest element in L, respectively. The set of all L-fuzzy natural integers is denoted by N(L). Definition 2 (see [11]). LetWA© be a fuzzy set, and ªdefine a map |A| : N → [0, 1] a ∈ (0, 1] : |A[a] | > n . Then |A| ∈ N([0, 1]), which such that ∀n ∈ N, |A|(n) = is called the cardinality of A. Definition 3 (see [10]). For any λ, µ ∈ N([0, 1]), the addition λ + µ of λ and µ is defined as follows: for any n ∈ N, _ (λ(k) ∧ µ(l)) . (λ + µ)(n) = k+l=n
Theorem 1 (see [10]). For any λ, µ ∈ N([0, 1]) and for any a ∈ [0, 1), one has (λ + µ)(a) = λ(a) + µ(a) . e = (E, µ), where E is a Definition 4 (see [4]). A fuzzy vector space is a pair E vector space over a field F , and µ : E → [0, 1] is a mapping satisfying µ(kx + ly) ≥ µ(x) ∧ µ(y) for any x, y ∈ E, k, l ∈ F . e = (E, µ) is a fuzzy vector space, define If E e[a] = µ[a] = {x ∈ E : µ(x) > a}, E
e(a) = µ(a) = {x ∈ E : µ(x) > a}, E
e[a] and E e(a) are the subspaces of E [4]. then both E
Fuzzy bases and the fuzzy dimension
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e1 = (E, µ1 ) and E e2 = (E, µ2 ) be two fuzzy vector Definition 5 (see [7]). Let E e1 and E e2 are respectively defined to spaces on E. The intersection and the sum of E e1 ∩ E e2 = (E, µ1 ∧ µ2 ) and E e1 + E e2 = (E, µ1 + µ2 ), where µ1 ∧ µ2 and µ1 + µ2 be E are respectively defined by (µ1 ∧ µ2 )(x) = µ1 (x) W ∧ µ2 (x) (µ1 + µ2 )(x) = (µ1 (x1 ) ∧ µ2 (x2 )) x=x W 1 +x2 = (µ1 (x1 ) ∧ µ2 (x − x1 )). x1 ∈E
e1 ∩ E e2 and E e1 + E e2 are fuzzy vector spaces. From [7] we know that both E e1 = (E, µ1 ) and E e2 = (E, µ2 ), we have Theorem 2. For two fuzzy vector spaces E the following results: e1 ∩ E e2 )[a] = (E e1 )[a] ∩ (E e2 )[a] ; (i) For all a ∈ [0, 1], (E e1 ∩ E e2 )(a) = (E e1 )(a) ∩ (E e2 )(a) ; (ii) For all a ∈ [0, 1), (E e e e e2 )(a) . (iii) For any a ∈ [0, 1), (E1 + E2 )(a) = (E1 )(a) + (E Proof. By Definition 5, we can easily obtain (i) and (ii). Statement (iii) can be proved as follows. For any a ∈ [0, 1), we have e1 + E e2 )(a) ⇔ x ∈ (E
W x1 +x2 =x
(µ1 (x1 ) ∧ µ2 (x2 )) > a
⇔ ∃x1 , x2 such that x1 + x2 = x and µ1 (x1 ) ∧ µ2 (x2 ) > a ⇔ ∃x1 , x2 such that x1 + x2 = x, x1 ∈ (µ1 )(a) and x2 ∈ (µ2 )(a) e1 )(a) + (E e2 )(a) . ⇔ x ∈ (E
In this paper, we suppose that E is a finite dimensional vector space.
3. A new definition of fuzzy bases In this section, we shall present a new definition of fuzzy bases for fuzzy vector spaces. e = (E, µ) be a fuzzy vector space and dim E = n. Lowen [5] proved that Let E µ(E) is a finite subset of [0, 1]. It is easy to prove the following lemma. e = (E, µ) is a fuzzy vector space, then there exists a finite sequence Lemma 1. If E 1 = α0 ≥ α1 > α2 > · · · > αr ≥ 0 such that (i) If a, b ∈ (αi+1 , αi ], then µ[a] = µ[b] ; (ii) If a ∈ (αi+1 , αi ] and b ∈ (αi , αi−1 ], then µ[a] ) µ[b] ; (iii) If a, b ∈ [αi+1 , αi ), then µ(a) = µ(b) ; (iv) If a ∈ [αi+1 , αi ) and b ∈ [αi , αi−1 ), then µ(a) ) µ(b) .
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e = (E, µ), by Lemma 1 we can obtain a family of For a fuzzy vector space E vector spaces as follows: {0} ⊆ µ[α1 ] ( µ[α2 ] ( · · · ( µ[αr ] ⊆ E.
(2)
e = (E, µ). Suppose We shall call (2) the family of irreducible level subspaces of E that µ[α1 ] 6= {0}, otherwise we can choose µ[α2 ] . Let Bα1 be a basis of µ[α1 ] . We can obtain a basis Bα2 of µ[α2 ] by extending Bα1 . Further, we can obtain a basis Bα3 of µ[α3 ] by extending Bα2 . Analogously, we can obtain a basis Bαr of µ[αr ] by extending Bαr−1 . Thus we obtain a sequence Bα1 ( Bα2 ( Bα3 ( · · · ( Bαr ,
(3)
where Bαi is a basis of µ[αi ] (1 ≤ i ≤ r). Therefore, we can define a fuzzy subset β of E as follows. β(x) =
_
{αi : x ∈ Bαi } .
(4)
e corresponding to (3). Then β is called a fuzzy basis of E The proof of the following theorem is trivial. e = (E, µ) obtained by (3). Then the following Theorem 3. Let β be a fuzzy basis of E statements hold: (i) If a, b ∈ (αi+1 , αi ], then β[a] = β[b] = Bαi ; (ii) If a ∈ (αi+1 , αi ] and b ∈ (αi , αi−1 ], then β[a] ) β[b] ; (iii) If a, b ∈ [αi+1 , αi ), then β(a) = β(b) = Bαi+1 ; (iv) If a ∈ [αi+1 , αi ) and b ∈ [αi , αi−1 ), then β(a) ) β(b) . e = (E, µ) obtained by (3). Then the Corollary 1. Let β be a fuzzy basis of E following statements hold: (i) a ∈ (0, 1], β[a] is a basis of µ[a] ; (ii) a ∈ [0, 1), β(a) is a basis of µ(a) . From the above corollary, we can easily obtain the following. e = (E, µ) be a fuzzy vector space and let β1 and β2 be two fuzzy Corollary 2. Let E e Then the following statements hold: bases of E. (i) For any a ∈ (0, 1], |(β1 )[a] | = |(β2 )[a] |; (ii) For any a ∈ [0, 1), |(β1 )(a) | = |(β2 )(a) |; (iii) |β1 | = |β2 |.
Fuzzy bases and the fuzzy dimension
307
4. A new definition of fuzzy dimension In this section, we redefine the fuzzy dimension of fuzzy vector spaces. The cardinality of a crisp set A can be regarded as an increasing set of integers {0, 1, · · · , n}. Such a set is also mathematically equivalent to the integer n. For a crisp vector space, its dimension was defined by the cardinality of its bases. We can define analogously the fuzzy dimension of fuzzy vector spaces as follows. e = (E, µ) be a fuzzy vector space with a fuzzy basis β. Define Definition 6. Let E e = |β|. Then dim(E) e is called the fuzzy dimension of E e = (E, µ). dim(E) e = (E, µ) be a fuzzy vector space with a fuzzy basis β. Then Theorem 4. Let E _© ª e dim(E)(n) = a ∈ [0, 1) : |β(a) | ≥ n ³ ´ o _n e(a) ≥ n = a ∈ [0, 1) : dim E ³ ´ e(a) = |β(a) |. For any n ∈ N, let Proof. By Corollary 1, we know that dim E λ=
_
³ ´ e(a) ≥ n}. {a ∈ [0, 1) : dim E
Obviously, we have e λ ≤ dim(E)(n) =
_©
ª a ∈ (0, 1] : |β[a] | ≥ n .
e e e In order to show that λ ≥ dim(E)(n), suppose that dim(E)(n) 6= 0 and dim(E)(n) > b. Then there exists na > b such that |β[a] | ≥ n. In this case, n ≤ |β | ≤ |β | ≤ |β |. [a] (b) [b] o W e This implies λ = a ∈ [0, 1) : dim(E)(a) ≥ n ≥ b. Thus we have λ≥
o _n e e b : 0 ≤ b < dim(E)(n) = dim(E)(n).
This completes the proof. e = (E, µ) be a fuzzy vector space. Then Theorem 5. Let E ³ ´ e [a] = dim E e[a] ; (i) For any a ∈ (0, 1], (dim(E)) ³ ´ e (a) = dim E e(a) . (ii) For any a ∈ [0, 1), (dim(E)) Proof. Let {0} ⊆ µ[α1 ] ( µ[α2 ] ( · · · ( µ[αr ] ⊆ E e = (E, µ). be the family of irreducible level subspaces of E ´ ³ ´ ³ e e that dim E e[a] ≤ dim(E) . Now (i) We know from definition of dim(E) [a] ´ ³ ´ ³ e e[a] . From the definition of fuzzy we need to show that dim(E) ≤ dim E [a]
308 dimension, we have ³ ´ e n ≤ dim(E)
F. -G. Shi and C. -E Huang
[a]
e ⇒ dim(E)(n) ≥a W e[α ] ) ≥ n} ≥ a ⇒ {αi : dim(E i ³
´ e[α ] ⇒ ∃αi ≥ a such that n ≤ dim E ´ ³ ´ i ³ e e ⇒ n ≤ dim E[αi ] ≤ dim E[a] . ³ ´ ³ ´ e e[a] for any a ∈ (0, 1]. Therefore, dim(E) = dim E [a] ³ ´ ³ ´ e e(a) . We suppose that (ii) In order to prove dim(E) ≤ dim E (a)
³ ´ e n = dim(E)
(a)
.
³ ´ e[b] ≥ n} > a. Hence, there {b ∈ (0, 1] : dim E ³ ´ e[b] . Since E e[b] ⊆ E e(a) , thus exists b ∈ (0, 1] such that a < b and n ≤ dim E ³ ´ ³ ´ ³ ´ e(a) Therefore, dim(E) e e(a) . n ≤ dim E ≤ dim E ³ ´ ³ (a) ´ e(a) = µ[α ] . e(a) ≤ dim(E) e , take αi > a such that E In order to show dim E i e Then dim(E)(n) > a, i.e.,
W
(a)
Then it is easy to see that ³ ´ ´ ¡ ¢ ³ e(a) = dim µ[α ] ≤ dim(E) e dim E i
[αi ]
³ ´ e ≤ dim(E)
(a)
.
e1 = (E, µ1 ) and E e2 = (E, µ2 ) be two fuzzy vector spaces, then it Theorem 6. Let E holds e1 + E e2 ) + dim(E e1 ∩ E e2 ) = dim(E e1 ) + dim(E e2 ). dim(E Proof. For any a ∈ [0, 1), by Theorem 1, Theorem 2 and Theorem 5 we have ³ ´ ³ ´ ³ ´ e1 + E e2 ) + dim(E e1 ∩ E e2 ) e1 + E e2 ) e1 ∩ E e2 ) dim(E = dim(E + dim(E (a) (a) (a) µ³ ´ ¶ ³ ´ e1 + E e2 e1 ∩ E e2 )(a) = dim E + dim (E (a) ³ ´ e1 )(a) + (E e2 )(a) = dim (E ³ ´ e1 )(a) ∩ (E e2 )(a) + dim (E ³ ´ ³ ´ e1 )(a) + dim (E e2 )(a) = dim (E ³ ´ ³ ´ e1 ) e2 ) = dim(E + dim(E (a) (a) ³ ´ e1 ) + dim(E e2 ) = dim(E (a)
e1 + E e2 ) + dim(E e1 ∩ E e2 ) = dim(E e1 ) + dim(E e2 ). Therefore, dim(E
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Fuzzy bases and the fuzzy dimension
e = (E, µ) be a fuzzy vector space and f : E → E a Definition 7 (see [2]). Let E linear transformation. If for all x ∈ E, µ(f (x)) ≥ µ(x), then f is called a fuzzy e = (E, µ). linear transformation on E e = (E, µ) be a fuzzy vector space and let f be a fuzzy linear Lemma 2. Let E f = (ker f, µ|ker f ) and imf e Then kerf f = (imf, µ|imf ) are two transformation on E. fuzzy vector spaces. Proof. It is trivial and it is omitted. e = (E, µ) be a fuzzy vector space and let f : E → E be a fuzzy Theorem 7. Let E e Then linear transformation on E. f ) + dim(imf e f ) = dim(E) dim(kerf Proof. By Theorem 1 and Theorem 5 it holds that for all a ∈ [0, 1), ³ ´ ³ ´ ³ ´ f ) f ) f ) + dim(kerf f ) dim(imf = dim(imf + dim(kerf (a) (a) (a)´ ³ ´ ³ f )(a) f )(a) + dim (kerf = dim (imf ³ ´ ³ ´ e(a) ∩ imf + dim E e(a) ∩ ker f . = dim E e(a) . Thus we have It is easy to check that f |Ee(a) is a linear transformation on E ³
f ) f ) + dim(kerf dim(imf
´ (a)
= dim(imf |Ee(a) ) + dim(ker f |Ee(a) ) ³ ´ ³ ´ e(a) = dim(E) e . = dim E (a)
f ) + dim(imf e f ) = dim(E). Therefore, dim(kerf
Acknowledgement The authors would like to thank reviewers for their valuable suggestions.
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[7] P. Lubczonok, Fuzzy vector spaces, Fuzzy Sets and Systems 38(1990), 329–343. [8] D. Rocacher, P. Bosc, The set of fuzzy rational numbers and flexible querying, Fuzzy Sets and Systems 155(2005), 317–339. [9] F. -G. Shi, L-fuzzy relation and L-fuzzy subgroup, J. Fuzzy Mathematics 8(2000), 491– 499. [10] F. -G. Shi, A new approach to the fuzzification of matroids, Fuzzy Sets and Systems 160(2009), 696–705. [11] L. A. Zadeh, A computational approach to fuzzy quantifiers in natural languages, Comput. Math. Appl. 9(1983), 149–184.
